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# PROBLEMS - chapter 5

Module by: NGUYEN Phuc. E-mail the author

PROBLEMS

This lecture note is based on the textbook # 1. Electric Machinery - A.E. Fitzgerald, Charles Kingsley, Jr., Stephen D. Umans- 6th edition- Mc Graw Hill series in Electrical Engineering. Power and Energy

5.1 The full-load torque angle of a synchronous motor at rated voltage and frequency is 35 electrical degrees. Neglect the effects of armature resistance and leakage reactance. If the field current is held constant, how would the full-load torque angle be affected by the following changes in operating condition?

a. Frequency reduced 10 percent, load torque and applied voltage constant.

b. Frequency reduced 10 percent, load power and applied voltage constant.

c. Both frequency and applied voltage reduced 10 percent, load torque constant.

d. Both frequency and applied voltage reduced 10 percent, load power constant.

5.2 Design calculations show the following parameters for a three-phase, cylindrical-rotor synchronous generator:

Phase-a self-inductance LaaLaa size 12{L rSub { size 8{ ital "aa"} } } {} = 4.83 mH

Armature leakage inductance La1La1 size 12{L rSub { size 8{a1} } } {} = 0.33 mH

Calculate the phase-phase mutual inductance and the machine synchronous inductance.

5.3 The open-circuit terminal voltage of a three-phase, 60-Hz synchronous generator is found to be 15.4 kV rms line-to-line when the field current is 420 A.

a. Calculate the stator-to-rotor mutual inductance LafLaf size 12{L rSub { size 8{ ital "af"} } } {}.

b. Calculate the open-circuit terminal voltage if the field current is held constant while the generator speed is reduced so that the frequency of the generated voltage is 50 Hz.

5.4 A 460-V, 50-kW, 60-Hz, three-phase synchronous motor has a synchronous reactance of XS=4.15ΩXS=4.15Ω size 12{X rSub { size 8{S} } =4 "." "15" %OMEGA } {} and an armature-to-field mutual inductance, LafLaf size 12{L rSub { size 8{ ital "af"} } } {} = 83 mH. The motor is operating at rated terminal voltage and an input power of 40 kW. Calculate the magnitude and phase angle of the line-toneutral generated voltage EˆafEˆaf size 12{ { hat {E}} rSub { size 8{ ital "af"} } } {} and the field current IfIf size 12{I rSub { size 8{f} } } {} if the motor is operating at (a) 0.85 power factor lagging, (b) unity power factor, and (c) 0.85 power factor leading.

5.5 The motor of Problem 5.4 is supplied from a 460-V, three-phase source through a feeder whose impedance is ZfZf size 12{Z rSub { size 8{f} } } {} = 0.084 + j0.82 ΩΩ size 12{ %OMEGA } {}. Assuming the system (as measured at the source) to be operating at an input power of 40 kW, calculate the magnitude and phase angle of the line-to-neutral generated voltage EˆafEˆaf size 12{ { hat {E}} rSub { size 8{ ital "af"} } } {}and the field current IfIf size 12{I rSub { size 8{f} } } {} for power factors of (a) 0.85 lagging, (b) unity, and (c) 0.85 leading.

5.6 A 50-Hz, two-pole, 750 kVA, 2300 V, three-phase synchronous machine has a synchronous reactance of 7.75 ΩΩ size 12{ %OMEGA } {} and achieves rated open-circuit terminal voltage at a field current of 120 A.

a. Calculate the armature-to-field mutual inductance.

b. The machine is to be operated as a motor supplying a 600 kW load at its rated terminal voltage. Calculate the internal voltage EafEaf size 12{E rSub { size 8{ ital "af"} } } {} and the corresponding field current if the motor is operating at unity power factor.

c. For a constant load power of 600 kW, write a MATLAB script to plot the terminal current as a function of field current. For your plot, let the field current vary between a minimum value corresponding to a machine loading of 750 kVA at leading power factor and a maximum value corresponding to a machine loading of 750 kVA at lagging power factor. What value of field current produces the minimum terminal current? Why?

5.7 The manufacturer's data sheet for a 26-kV, 750-MVA, 60-Hz, three-phase synchronous generator indicates that it has a synchronous reactance XSXS size 12{X rSub { size 8{S} } } {} = 2.04 and a leakage reactance Xa1Xa1 size 12{X rSub { size 8{a1} } } {} = 0.18, both in per unit on the generator base. Calculate (a) the synchronous inductance in mH, (b) the armature leakage inductance in mH, and (c) the armature phase inductance LaaLaa size 12{L rSub { size 8{ ital "aa"} } } {} in mH and per unit.

5.8 The following readings are taken from the results of an open- and a shortcircuit test on an 800-MVA, three-phase, Y-connected, 26-kV, two-pole, 60-Hz turbine generator driven at synchronous speed:

The number in parentheses are extrapolations based upon the measured data. Find (a) the short-circuit ratio, (b) the unsaturated value of the synchronous reactance in ohms per phase and per unit, and (c) the saturated synchronous reactance in per unit and in ohms per phase.

5.9 The following readings are taken from the results of an open- and a short-circuit test on a 5000-kW, 4160-V, three-phase, four-pole, 1800-rpm synchronous motor driven at rated speed:

The armature resistance is 11 m ΩΩ size 12{ %OMEGA } {}/phase. The armature leakage reactance is estimated to be 0.12 per unit on the motor rating as base. Find (a) the short-circuit ratio, (b) the unsaturated value of the synchronous reactance in ohms per phase and per unit, and (c) the saturated synchronous reactance in per unit and in ohms per phase.

5.10 Write a MATLAB script which automates the calculations of Problems 5.8 and 5.9. The following minimum set of data is required:

• AFNL: The field current required to achieve rated open-circuit terminal voltage.
• The corresponding terminal voltage on the air gap line.
• AFSC: The field current required to achieve rated short-circuit current on the short-circuit characteristic.

Your script should calculate (a) the short-circuit ratio, (b) the unsaturated value of the synchronous reactance in ohms per phase and per unit, and (c) the saturated synchronous reactance in per unit and in ohms per phase.

5.11 Consider the motor of Problem 5.9.

a. Compute the field current required when the motor is operating at rated voltage, 4200 kW input power at 0.87 power factor leading. Account for saturation under load by the method described in the paragraph relating to Eq. Xs=Va,ratedIa'Xs=Va,ratedIa' size 12{X rSub { size 8{s} } = { {V rSub { size 8{a, ital "rated"} } } over { { {I}} sup { ' } rSub { size 8{a} } } } } {}.

b. In addition to the data given in Problem 5.9, additional points on the open-circuit characteristic are given below:

If the circuit breaker supplying the motor of part (a) is tripped, leaving the motor suddenly open-circuited, estimate the value of the motor terminal voltage following the trip (before the motor begins to slow down and before any protection circuitry reduces the field current).

5.12 Using MATLAB, plot the field current required to achieve unity-power-factor operation for the motor of Problem 5.9 as the motor load varies from zero to full load. Assume the motor to be operating at rated terminal voltage.

5.13 Loss data for the motor of Problem 5.9 are as follows:

Open-circuit core loss at 4160 V = 37 kW

Friction and windage loss = 46 kW

Field-winding resistance at 750C750C size 12{"75" rSup { size 8{0} } C} {} = 0.279 ΩΩ size 12{ %OMEGA } {}

Compute the output power and efficiency when'the motor is operating at rated input power, unity power factor, and rated voltage. Assume the field-winding to be operating at a temperature of 1250C1250C size 12{"125" rSup { size 8{0} } C} {}.

5.14 The following data are obtained from tests on a 145-MVA, 13.8-kV, threephase, 60-Hz, 72-pole hydroelectric generator. Open-circuit characteristic:

Short-circuit test:

IfIf size 12{I rSub { size 8{f} } } {} = 710 A, IaIa size 12{I rSub { size 8{a} } } {} = 6070 A

a. Draw (or plot using MATLAB) the open-circuit saturation curve, the air-gap line, and the short-circuit characteristic.

b. Find AFNL and AFSC. (Note that if you use MATLAB for part (a), you can use the MATLAB function 'polyfit' to fit a second-order polynomial to the open-circuit saturation curve. You can then use this fit to find AFNL.)

c. Find (i) the short-circuit ratio, (ii) the unsaturated value of the synchronous reactance in ohms per phase and per unit and (iii) the saturated synchronous reactance in per unit and in ohms per phase.

5.15 What is the maximum per-unit reactive power that can be supplied by a synchronous machine operating at its rated terminal voltage whose synchronous reactance is 1.6 per unit and whose maximum field current is limited to 2.4 times that required to achieve rated terminal voltage under open-circuit conditions?

5.16 A 25-MVA, 11.5 kV synchronous machine is operating as a synchronous condenser, as discussed in Appendix D (section D.4.1). The generator short-circuit ratio is 1.68 and the field current at rated voltage, no load is 420 A. Assume the generator to be connected directly to an 11.5 kV source.

a. What is the saturated synchronous reactance of the generator in per unit and in ohms per phase?

The generator field current is adjusted to 150 A.

b. Draw a phasor diagram, indicating the terminal voltage, internal voltage, and armature current.

c. Calculate the armature current magnitude (per unit and amperes) and its relative phase angle with respect to the terminal voltage.

d. Under these conditions, does the synchronous condenser appear inductive or capacitive to the 11.5 kV system?

e. Repeat parts (b) through (d) for a field current of 700 A.

5.17 The synchronous condenser of Problem 5.16 is connected to a 11.5 kV system through a feeder whose series reactance is 0.12 per unit on the machine base. Using MATLAB, plot the voltage (kV) at the synchronous-condenser terminals as the synchronous-condenser field current is varied between 150 A and 700 A.

5.18 A synchronous machine with a synchronous reactance of 1.28 per unit is operating as a generator at a real power loading of 0.6 per unit connected to a system with a series reactance of 0.07 per unit. An increase in its field current is observed to cause a decrease in armature current.

a. Before the increase, was the generator supplying or absorbing reactive power from the power system?

b. As a result of this increase in excitation, did the generator terminal voltage increase or decrease?

c. Repeat parts (a) and (b) if the synchronous machine is operating as a motor.

5.19 Superconducting synchronous machines are designed with superconducting fields windings which can support large current densities and create large magnetic flux densities. Since typical operating magnetic flux densities exceed the saturation flux densities of iron, these machines are typically designed without iron in the magnetic circuit; as a result, these machines exhibit no saturation effects and have low synchronous reactances.

Consider a two-pole, 60-Hz, 13.8-kV, 10-MVA superconducting generator which achieves rated open-circuit armature voltage at a field current of 842 A. It achieves rated armature current into a three-phase terminal short circuit for a field current of 226 A.

a. Calculate the per-unit synchronous reactance. Consider the situation in which this generator is connected to a 13.8 kV distribution feeder of negligible impedance and operating at an output power of 8.75 MW at 0.9 pf lagging. Calculate:

b. the field current in amperes, the reactive-power output in MVA, and the rotor angle for this operating condition.

c. the resultant rotor angle and reactive-power output in MVA if the field current is reduced to 842 A while the shaft-power supplied by the prime mover to the generator remains constant.

5.20For a synchronous machine with constant synchronous reactance XSXS size 12{X rSub { size 8{S} } } {} operating at a constant terminal voltage VtVt size 12{V rSub { size 8{t} } } {} and a constant excitation voltage EafEaf size 12{E rSub { size 8{ ital "af"} } } {}, show that the locus of the tip of the armature-current phasor is a circle. On a phasor diagram with terminal voltage shown as the reference phasor, indicate the position of the center of this circle and its radius. Express the coordinates of the center and the radius of the circle in terms of Vt,Eaf,XSVt,Eaf,XS size 12{V rSub { size 8{t} } ,E rSub { size 8{ ital "af"} } ,X rSub { size 8{S} } } {}.

5.21 A four-pole, 60-Hz, 24-kV, 650-MVA synchronous generator with a synchronous reactance of 1.82 per unit is operating on a power system which can be represented by a 24-kV infinite bus in series with a reactive impedance of j0.21 ΩΩ size 12{ %OMEGA } {}. The generator is equipped with a voltage regulator that adjusts the field excitation such that the generator terminal voltage remains at 24 kV independent of the generator loading.

a. The generator output power is adjusted to 375 MW.

(i) Draw a phasor diagram for this operating condition.

(ii) Find the magnitude (in kA) and phase angle (with respect to the generator terminal voltage) of the terminal current.

(iii) Determine the generator terminal power factor.

(iv) Find the magnitude (in per unit and kV) of the generator excitation voltage EafEaf size 12{E rSub { size 8{ ital "af"} } } {}.

b. Repeat part (a) if the generator output power is increased to 600 MW.

5.22 The generator of Problem 5.21 achieves rated open-circuit armature voltage at a field current of 850 A. It is operating on the system of Problem 5.21 with its voltage regulator set to maintain the terminal voltage at 0.99 per unit (23.8 kV).

a. Use MATLAB to plot the generator field current (in A) as a function of load (in MW) as the load on the generator output power is varied from zero to full load.

b. Plot the corresponding reactive output power in MVAR as a function of output load.

c. Repeat the plots of parts (a) and (b) if the voltage regulator is set to regulate the terminal voltage to 1.01 per unit (24.2 kV).

5.23 The 145 MW hydroelectric generator of Problem 5.14 is operating on a 13.8-kV power system. Under normal operating procedures, the generator is operated under automatic voltage regulation set to maintain its terminal voltage at 13.8 kV. In this problem you will investigate the possible consequences should the operator forget to switch over to the automatic voltage regulator and instead leave the field excitation constant at AFNL, the value corresponding to rated open-circuit voltage. For the purposes of this problem, neglect the effects of saliency and assume that the generator can be represented by the saturated synchronous reactance found in Problem 5.14.

a. If the power system is represented simply by a 13.8 kV infinite (ignoring the effects of any equivalent impedance), can the generator be loaded to full load? If so, what is the power angle δδ size 12{δ} {} corresponding to full load? If not, what is the maximum load that can be achieved?

b. Repeat part (a) with the power system now represented by a 13.8 kV infinite bus in series with a reactive impedance of j0.14 ΩΩ size 12{ %OMEGA } {}.

5.24 Repeat Problem 5.23 assuming that the saturated direct-axis synchronous inductance XdXd size 12{X rSub { size 8{d} } } {} is equal to that found in Problem 5.14 and that the saturated quadrature-axis synchronous reactance XqXq size 12{X rSub { size 8{q} } } {} is equal to 75 percent of this value. Compare your answers to those found in Problem 5.23.

5.25 Write a MATLAB script to plot a set of per-unit power-angle curves for a salient-pole synchronous generator connected to an infinite bus ( VbusVbus size 12{V rSub { size 8{ ital "bus"} } } {} = 1.0 per unit). The generator reactances are XdXd size 12{X rSub { size 8{d} } } {} = 1.27 per unit and XqXq size 12{X rSub { size 8{q} } } {} = 0.95 per unit. Assuming EafEaf size 12{E rSub { size 8{ ital "af"} } } {} = 1.0 per unit, plot the following curves:

a. Generator connected directly to the infinite bus.

b. Generator connected to the infinite bus through a reactance XbusXbus size 12{X rSub { size 8{ ital "bus"} } } {} = 0.1 per unit.

c. Generator connected directly to the infinite bus. Neglect saliency effects, setting Xd=XqXd=Xq size 12{X rSub { size 8{d} } =X rSub { size 8{q} } } {}.

d. Generator connected to the infinite bus through a reactance XbusXbus size 12{X rSub { size 8{ ital "bus"} } } {} = 0.1 per unit. Neglect saliency effects, setting Xq=XdXq=Xd size 12{X rSub { size 8{q} } =X rSub { size 8{d} } } {}.

5.26 Draw the steady-state, direct- and quadrature-axis phasor diagram for a salient-pole synchronous motor with reactances XdXd size 12{X rSub { size 8{d} } } {} and XqXq size 12{X rSub { size 8{q} } } {} and armature resistance RaRa size 12{R rSub { size 8{a} } } {}. From this phasor diagram, show that the torque angle δ between the generated voltage EˆafEˆaf size 12{ { hat {E}} rSub { size 8{ ital "af"} } } {} (which lies along the quadrature axis) and the terminal voltage VˆtVˆt size 12{ { hat {V}} rSub { size 8{t} } } {} is given by

tan δ = I a X q cos φ + I a R a sin φ V t + I a X q sin φ I a R a cos φ tan δ = I a X q cos φ + I a R a sin φ V t + I a X q sin φ I a R a cos φ size 12{"tan"δ= { {I rSub { size 8{a} } X rSub { size 8{q} } "cos"φ+I rSub { size 8{a} } R rSub { size 8{a} } "sin"φ} over {V rSub { size 8{t} } +I rSub { size 8{a} } X rSub { size 8{q} } "sin"φ - I rSub { size 8{a} } R rSub { size 8{a} } "cos"φ} } } {}

Here φφ size 12{φ} {} is the phase angle of the armature current IˆaIˆa size 12{ { hat {I}} rSub { size 8{a} } } {} and VtVt size 12{V rSub { size 8{t} } } {}, considered to be negative when IˆaIˆa size 12{ { hat {I}} rSub { size 8{a} } } {} lags VˆtVˆt size 12{ { hat {V}} rSub { size 8{t} } } {}.

5.27 Repeat Problem 5.26 for synchronous generator operation, in which case the equation for δδ size 12{δ} {} becomes

tan δ = I a X q cos φ + I a R a sin φ V t I a X q sin φ + I a R a cos φ tan δ = I a X q cos φ + I a R a sin φ V t I a X q sin φ + I a R a cos φ size 12{"tan"δ= { {I rSub { size 8{a} } X rSub { size 8{q} } "cos"φ+I rSub { size 8{a} } R rSub { size 8{a} } "sin"φ} over {V rSub { size 8{t} } - I rSub { size 8{a} } X rSub { size 8{q} } "sin"φ+I rSub { size 8{a} } R rSub { size 8{a} } "cos"φ} } } {}

5.28 What maximum percentage of its rated output power will a salient-pole motor deliver without loss of synchronism when operating at its rated terminal voltage with zero field excitation ( ***SORRY, THIS MEDIA TYPE IS NOT SUPPORTED.*** = 0) if XdXd size 12{X rSub { size 8{d} } } {} = 0.90 per unit and XqXq size 12{X rSub { size 8{q} } } {} = 0.65 per unit? Compute the per-unit armature current and reactive power for this operating condition.

5.29 If the synchronous motor of Problem 5.28 is now operated as a synchronous generator connected to an infinite bus of rated voltage, find the minimum per-unit field excitation (where 1.0 per unit is the field current required to achieve rated open-circuit voltage) for which the generator will remain synchronized at (a) half load and (b) full load.

5.30 A salient-pole synchronous generator with saturated synchronous reactances ***SORRY, THIS MEDIA TYPE IS NOT SUPPORTED.*** =1.57 per unit and XqXq size 12{X rSub { size 8{q} } } {} = 1.34 per unit is connected to an infinite bus of rated voltage through an external impedance XbusXbus size 12{X rSub { size 8{ ital "bus"} } } {} = 0.11 per unit. The generator is supplying its rated MVA at 0.95 power factor lagging, as measured at the generator terminals.

a. Draw a phasor diagram, indicating the infinite-bus voltage, the armature current, the generator terminal voltage, the excitation voltage, and the rotor angle (measured with respect to the infinite bus).

b. Calculate the per-unit terminal and excitation voltages, and the rotor angle in degrees.

5.31 A salient-pole synchronous generator with saturated synchronous reactances XdXd size 12{X rSub { size 8{d} } } {} = 0.78 per unit and XqXq size 12{X rSub { size 8{q} } } {} = 0.63 per unit is connected to a rated-voltage infinite bus through an external impedance XbusXbus size 12{X rSub { size 8{ ital "bus"} } } {} = 0.09 per unit.

a. Assuming the generator to be supplying only reactive power

(i) Find minimum and maximum per-unit field excitations (where 1.0 per unit is the field current required to achieve rated open-circuit voltage) such that the generator does not exceed its rated terminal current.

(ii) Using MATLAB, plot the armature current as a function of field excitation as the field excitation is varied between the limits determined in part (i).

b. Now assuming the generator to be supplying 0.25 per unit rated real power, on the same axes add a plot of the per-unit armature current as a function of field excitation as the field current is varied in the range for which the per-unit armature current is less than 1.0 per unit.

c. Repeat part (b) for generator output powers of 0.50 and 0.75 per unit. The final result will be a plot of V-curves for this generator in this configuration.

5.32 A two-phase permanent-magnet ac motor has a rated speed of 3000 r/min and a six-pole rotor. Calculate the frequency (in Hz) of the armature voltage required to operate at this speed.

5.33 A 5-kW, three-phase, permanent-magnet synchronous generator produces an open-circuit voltage of 208 V line-to-line, 60-Hz, when driven at a speed of 1800 r/min. When operating at rated speed and supplying a resistive load, its terminal voltage is observed to be 192 V line-to-line for a power output of 4.5 kW.

a. Calculate the generator phase current under this operating condition.

b. Assuming the generator armature resistance to be negligible, calculate the generator 60-Hz synchronous reactance.

c. Calculate the generator terminal voltage which will result if the motor generator load is increased to 5 kW (again purely resistive) while the speed is maintained at 1800 r/min.

5.34 Small single-phase permanent-magnet ac generators are frequently used to generate the power for lights on bicycles. For this application, these generators are typically designed with a significant amount of leakage inductance in their armature winding. A simple model for these generators is an ac voltage source ea(t)=ωKacosωtea(t)=ωKacosωt size 12{e rSub { size 8{a} } $$t$$ =ωK rSub { size 8{a} } "cos"ωt} {} in series with the armature leakage inductance La and the armature resistance RaRa size 12{R rSub { size 8{a} } } {}. Here ω is the electrical frequency of the generated voltage which is determined by the speed of the generator as it rubs against the bicycle wheel.

Assuming that the generator is running a light bulb which can be modeled as a resistance RbRb size 12{R rSub { size 8{b} } } {}, write an expression for the minimum frequency ωminωmin size 12{ω rSub { size 8{"min"} } } {} which must be achieved in order to insure that the light operates at constant brightness, independent of the speed of the bicycle.

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